Large Language Models (LLMs) have demonstrated remarkable performance across various natural language processing (NLP) tasks. However, their deployment is challenging due to the substantial computational resources required. Power-of-two (PoT) quantization is a general tool to counteract this difficulty. Albeit previous works on PoT quantization can be efficiently dequantized on CPUs using fixed-point addition, it showed less effectiveness on GPUs. The reason is entanglement of the sign bit and sequential bit manipulations needed for dequantization. We propose a novel POT quantization framework for LLM weights that (i) outperforms state-of-the-art accuracy in extremely low-precision number formats, and (ii) enables faster inference through more efficient dequantization. To maintain the accuracy of the quantized model, we introduce a two-step post-training algorithm: (i) initialize the quantization scales with a robust starting point, and (ii) refine these scales using a minimal calibration set. The performance of our PoT post-training algorithm surpasses the current state-of-the-art in integer quantization, particularly at low precisions such as 2- and 3-bit formats. Our PoT quantization accelerates the dequantization step required for the floating point inference and leads to 3.67× speed up on a NVIDIA V100, and 1.63× on a NVIDIA RTX 4090, compared to uniform integer dequantization.

Suyuchen Wang, Jinlin Wang, Xinyu Wang, Shiqi Li, Xiangru Tang, Sirui Hong, Xiao-Wen Chang, Chenglin Wu, Bang Liu. Proceedings of the 2025 Conference on Empirical Methods in Natural Language Processing. 2025.

Massive MIMO (multiple-input multiple-output) detection is an important topic in wireless communication and various machine learning based methods have been developed recently for this task. Expectation propagation (EP) and its variants are widely used for MIMO detection and have achieved the best performance. However, EP-based solvers fail to capture the correlation between unknown variables, leading to a loss of information, and in addition, they are computationally expensive. In this paper, we show that the real-valued system can be modeled as spectral signal convolution on graph, through which the correlation between unknown variables can be captured. Based on this analysis, we propose graph convolution-enhanced expectation propagation (GCEPNet). GCEPNet incorporates data-dependent attention scores into Chebyshev polynomial for powerful graph convolution with better generalization capacity. It enables a better estimation of the cavity distribution for EP and empirically achieves the state-of-the-art (SOTA) MIMO detection performance with much faster inference speed. To our knowledge, we are the first to shed light on the connection between the system model and graph convolution, and the first to design the data-dependent coefficients for graph convolution.

We consider the success probability of the$L_{0}{-}$regularized box-constrained Babai point, which is a suboptimal solution to the$L_{0}$-regularized box-constrained integer least squares problem and can be used for MIMO detection. First, we derive formulas for the success probability of both$L_{0}$-regularized and unregularized box-constrained Babai points. Then we investigate the properties of the$L_{0}$-regularized box-constrained Babai point, including the optimality of the regularization parameter and the monotonicity of the ratio of the two success probabilities. Finally a bound on the success probability of the$L_{0}$-regularized Babai point is derived.

This paper is concerned with success probabilities of some L2-norm regularized Babai detectors for MIMO detection. First, we extend the often used MMSE-SIC detector to two L2-norm regularized Babai detectors. Then, we derive the corresponding success probability formulae of the two detectors. To increase their success probabilities, we propose to maximize them with respect to the regularization parameter vector by a machine learning based method called ML-λ, as conventional optimization methods are very inefficient. This moves the exponential computational cost to offline training, and the method outputs an approximation to the optimal regularization parameter vector in polynomial time. Finally, we present some numerical simulations to demonstrate the performance of the new detectors.

Homophily principle, i.e., nodes with the same labels are more likely to be connected, has been believed to be the main reason for the performance superiority of Graph Neural Networks (GNNs) over Neural Networks on node classification tasks. Recent research suggests that, even in the absence of homophily, the advantage of GNNs still exists as long as nodes from the same class share similar neighborhood patterns. However, this argument only considers intra-class Node Distinguishability (ND) but neglects inter-class ND, which provides incomplete understanding of homophily on GNNs. In this paper, we first demonstrate such deficiency with examples and argue that an ideal situation for ND is to have smaller intra-class ND than inter-class ND. To formulate this idea and study ND deeply, we propose Contextual Stochastic Block Model for Homophily (CSBM-H) and define two metrics, Probabilistic Bayes Error (PBE) and negative generalized Jeffreys divergence, to quantify ND. With the metrics, we visualize and analyze how graph filters, node degree distributions and class variances influence ND, and investigate the combined effect of intra- and inter-class ND. Besides, we discovered the mid-homophily pitfall, which occurs widely in graph datasets. Furthermore, we verified that, in real-work tasks, the superiority of GNNs is indeed closely related to both intra- and inter-class ND regardless of homophily levels. Grounded in this observation, we propose a new hypothesis-testing based performance metric beyond homophily, which is non-linear, feature-based and can provide statistical threshold value for GNNs' the superiority. Experiments indicate that it is significantly more effective than the existing homophily metrics on revealing the advantage and disadvantage of graph-aware modes on both synthetic and benchmark real-world datasets.

Graph Neural Networks (GNNs) extend basic Neural Networks (NNs) by using graph structures based on the relational inductive bias (homophily assumption). While GNNs have been commonly believed to outperform NNs in real-world tasks, recent work has identified a non-trivial set of datasets where their performance compared to NNs is not satisfactory. Heterophily has been considered the main cause of this empirical observation and numerous works have been put forward to address it. In this paper, we first revisit the widely used homophily metrics and point out that their consideration of only graph-label consistency is a shortcoming. Then, we study heterophily from the perspective of post-aggregation node similarity and define new homophily metrics, which are potentially advantageous compared to existing ones. Based on this investigation, we prove that some harmful cases of heterophily can be effectively addressed by local diversification operation. Then, we propose the Adaptive Channel Mixing (ACM), a framework to adaptively exploit aggregation, diversification and identity channels to extract richer localized information in each baseline GNN layer. ACM is more powerful than the commonly used uni-channel framework for node classification tasks on heterophilic graphs. When evaluated on 10 benchmark node classification tasks, ACM-augmented baselines consistently achieve significant performance gain, exceeding state-of-the-art GNNs on most tasks without incurring significant computational burden. (Code: https://github.com/SitaoLuan/ACM-GNN)

We study the problem of direction of arrival (DoA) estimation for cell-free massive MIMO (m-MIMO) systems operating over extremely high frequency (EHF) and terahertz (THF) bands, where the wireless channel can effectively be modeled by a line-of-sight path. For this model, a low-complexity deep neural network (DNN)-based method is proposed to estimate the DoA of a radio wave impinging on an access point (AP) equipped with an antenna array. To train the DNN, a special feature set is proposed obtained from the first superdiagonal entries of the spatial correlation matrix. This selection of features makes it possible to employ a DNN with only a few low-dimensional layers, which considerably speeds up training and processing. More importantly, it is shown that the trained DNN is robust against quantization noise in the array snapshot data. This property makes the centralized implementation of the proposed DNN-based method feasible, which is particularly well-suited for cell-free m-MIMO. Through extensive simulations, the new method is shown to achieve an estimation performance that nearly matches or exceeds that of classical bechmark methods, but with considerably reduced complexity.

The current thin film transistor liquid crystal display (TFT-LCD) panel industry evolves in transition from volume production to customized production service as a major competition strategy. Owing to the highly mixed production type of small batch sizes and high varieties, metrology data are usually collected too infrequently for building accurate virtual metrology (VM) models. In this paper, a new product-to-product (P2P) VM model to predict photoresist spacer heights in the color filter process of the array sector is developed. First, a position-based random forests model is proposed for the base product. Second, an ensemble model combining the base model and a biased estimation random forests model for the new product of small batch size is proposed to perform P2P VM between different products on the same tool. The real datasets of photoresist spacer heights for different products are used to illustrate the accuracy of the new P2P VM framework in the experimental study. Note to Practitioners—As global competition intensifies, production management needs to be adjusted to adapt swiftly to changes in the time-varying and competitive market. For manufacturing firms, how to operate the manufacturing system effectively and efficiently plays a crucial role in quality and yield improvements. The color filter process in the array sector is of critical importance in the TFT-LCD practice since it essentially dictates the quality level of subsequent processing steps in the cell and assembly sectors. Virtual metrology is a state-of-the-art measure that can be used for monitoring and controlling purposes to maintain the yield and productivity of the color filter process in the highly mixed production environment. An immediate challenge to be faced by the data scientists is how to build accurate virtual metrology models for a variety of product lines separately in color filter manufacturing. A new proposal of product-to-product virtual metrology is first addressed between products, enabling to build appropriate virtual metrology models for new products with small-sized production batches.

In this paper, we propose a novel power allocation scheme for physical-layer network coding (PNC) in downlink multi-way relay channels (MWRC). The power allocation is formulated as a constrained optimization problem, where the aim is to maximize the success probability under a total power constraint when using Babai estimation for signal detection. Optimizing over this metric allows us to maximize the probability of successfully decoding a chain of network codes, which is of crucial importance in downlink multi-way PNC. Specifically, to meet the different requirements for transmission quality in applications, we consider different aggregate measures of success probability over the participating user terminals, i.e., the arithmetic mean, the geometric mean, and the maximin. For each measure, we formulate a constrained optimization and demonstrate the concavity of the objective, allowing us to obtain solution efficiently via iterative means. The performance of the proposed power allocation schemes for downlink PNC in MWRC is evaluated by means of computer simulations over Raylegih fading channels. The results demonstrate the effectiveness of the proposed schemes in improving the success probability in the reception of a chain of network codes.

This paper is concerned with detecting an integer parameter vector inside a box from a linear model that is corrupted with a noise vector following the Gaussian distribution. One of the commonly used detectors is the maximum likelihood detector, which is obtained by solving a box-constrained integer least squares problem, that is NP-hard. Two other popular detectors are the box-constrained rounding and Babai detectors due to their high efficiency of implementation. In this paper, we first present formulas for the success probabilities (the probabilities of correct detection) of these three detectors for two different situations: the integer parameter vector is deterministic and is uniformly distributed over the constraint box. Then, we give two simple examples to respectively show that the success probability of the box-constrained rounding detector can be larger than that of the box-constrained Babai detector and the latter can be larger than the success probability of the maximum likelihood detector when the parameter vector is deterministic, and prove that the success probability of the box-constrained rounding detector is always not larger than that of the box-constrained Babai detector when the parameter vector is uniformly distributed over the constraint box. Some relations between the results for the box constrained and ordinary cases are presented, and two bounds on the success probability of the maximum likelihood detector, which can easily be computed, are developed. Finally, simulation results are provided to illustrate our main theoretical findings.

Estimating the integer parameter vector in a linear model with additive Gaussian noise arises from many applications, including communications. The optimal approach is to solve an integer least squares (ILS) problem, which is unfortunately NP-hard. Recently Klein's randomized algorithm, which finds a sub-optimal solution to the ILS problem, to be referred to as the randomized Babai point, has attracted much attention. This paper presents a formula of the success probability of the randomized Babai point and some interesting properties, and compares it with the deterministic Babai point.

The Korkine-Zolotareff (KZ) reduction is a widely used lattice reduction strategy in communications and cryptography. The Hermite constant, which is a vital constant of lattice, has many applications, such as bounding the length of the shortest nonzero lattice vector and orthogonality defect of lattices. The KZ constant can be used in quantifying some useful properties of KZ reduced matrices. In this paper, we first develop a linear upper bound on the Hermite constant and then use the bound to develop an upper bound on the KZ constant. These upper bounds are sharper than those obtained recently by the first two authors. Some examples on the applications of the improved upper bounds are also presented.

Recently, neural network based approaches have achieved significant progress for solving large, complex, graph-structured problems. Nevertheless, the advantages of multi-scale information and deep architectures have not been sufficiently exploited. In this paper, we first analyze key factors constraining the expressive power of existing Graph Convolutional Networks (GCNs), including the activation function and shallow learning mechanisms. Then, we generalize spectral graph convolution and deep GCN in block Krylov subspace forms, upon which we devise two architectures, both scalable in depth however making use of multi-scale information differently. On several node classification tasks, the proposed architectures achieve state-of-the-art performance.

The Korkine-Zolotareff (KZ) reduction is one of the often used reduction strategies for lattice decoding. In this paper, we first investigate some important properties of KZ reduced matrices. Specifically, we present a linear upper bound on the Hermit constant which is around 7/8 times of the existing sharpest linear upper bound, and an upper bound on the KZ constant which is polynomially smaller than the existing sharpest one. We also propose upper bounds on the lengths of the columns of KZ reduced matrices, and an upper bound on the orthogonality defect of KZ reduced matrices which are even polynomially and exponentially smaller than those of boosted KZ reduced matrices, respectively. Then, we derive upper bounds on the magnitudes of the entries of any solution of a shortest vector problem (SVP) when its basis matrix is LLL reduced. These upper bounds are useful for analyzing the complexity and understanding numerical stability of the basis expansion in a KZ reduction algorithm. Finally, we propose a new KZ reduction algorithm by modifying the commonly used Schnorr-Euchner search strategy for solving SVPs and the basis expansion method proposed by Zhang et al. Simulation results show that the new KZ reduction algorithm is much faster and more numerically reliable than the KZ reduction algorithm proposed by Zhang et al., especially when the basis matrix is ill conditioned.

A general formula for the success probability of the regularized Babai point, which serves as a suboptimal solution to the sparse maximum a posteriori detection, is derived. As a special case, a formula for the success probability of the Babai point of the maximum likelihood detection is obtained. It is shown that the former is higher than the latter. Based on the general formula, a greedy column permutation strategy is proposed to enhance the success probability of the regularized Babai point.

In communications, one frequently needs to detect a parameter vector x in a box from a linear model. The box-constrained rounding detector xBRand Babai detector xBBare often used to detect x due to their high probability of correct detection, which is referred to as success probability, and their high efficiency of implimentation. It is generally believed that the success probability PBRof xBRis not larger than the success probability PBBof xBB. In this paper, we first present formulas for PBRand PBBfor two different situations: x is deterministic and x is uniformly distributed over the constraint box. Then, we give a simple example to show that PBRmay be strictly larger than PBBif x is deterministic, while we rigorously show that pBR≤ pBBalways holds if x is uniformly distributed over the constraint box.

State-of-the-art indoor localization techniques can reach high localization accuracy, but rely on widely deployed infrastructure which may be absent in less developed regions. To achieve infrastructure-free indoor localization, researchers have proposed to use the inertial sensors on the mobile devices to update the users potential locations by walking steps. Such methods are known to accumulate errors and degrade drastically over time. To compensate for it, either a wall-constraint that eliminates possible steps going through the walls, or landmarks correspond to the special user activities, are usually employed. However, these approaches still suffer from slow convergence and cannot detect floor changes automatically without the information from other users. In this paper, we propose LocMe, an indoor localization service based on human locomotion detection and map exploitation. By synergizing both the locomotion-constraint and wall-constraint, LocMe can significantly increase the converging speed of localization and automatically detect the floor changes, with negligible extra complexity. Our field tests show that LocMe can achieve a median localization error of 1.1 m, which is over 68% lower than the localization algorithm with only the wall-constraint in the same test condition.

In many applications including communications, one may encounter a linear model where the parameter vector x̂ is an integer vector in a box. To estimate x̂, a typical method is to solve a box-constrained integer least squares problem. However, due to its high complexity, the box-constrained Babai integer point xBBis commonly used as a suboptimal solution. In this paper, we first derive formulas for the success probability PBBof xBBand the success probability POB of the ordinary Babai integer point xOBwhen x̂ is uniformly distributed over the constraint box. Some properties of PBBand POBand the relationship between them are studied. Then, we investigate the effects of some column permutation strategies on PBB. In addition to V-BLAST and SQRD, we also consider the permutation strategy involved in the LLL lattice reduction, to be referred to as LLL-P. On the one hand, we show that when the noise is relatively small, LLL-P always increases PBBand argue why both V-BLAST and SQRD often increase PBB; and on the other hand, we show that when the noise is relatively large, LLL-P always decreases PBBand argue why both V-BLAST and SQRD often decrease PBB. We also derive a column permutation invariant bound on PBB, which is an upper bound and a lower bound under these two opposite conditions, respectively. Numerical results demonstrate our findings. Finally, we consider a conjecture concerning xOBproposed by Ma et al. We first construct an example to show that the conjecture does not hold in general, and then show that it does hold under some conditions.

We modify the algorithm proposed by Sahraei et al. in 2015, resulting an algorithm with expected complexity of O(n log n) arithmetic operations to solve a special shortest vector problem arising in computer-and-forward design, where n is the dimension of the channel vector. This algorithm is more efficient than the best known algorithms with proved complexity.

We consider the problem of finding the optimal coefficient vector that maximizes the computation rate at a relay in the compute-and-forward scheme. Based on the idea of sphere decoding, we propose a highly efficient algorithm that finds the optimal coefficient vector. First, we derive a novel algorithm to transform the original quadratic form optimization problem into a shortest vector problem (SVP) using the Cholesky factorization. Instead of computing the Cholesky factor explicitly, the proposed algorithm realizes the Cholesky factorization with only O(n) flops by taking advantage of the structure of the Gram matrix in the quadratic form. Then, we propose some conditions that can be checked with O(n) flops, under which a unit vector is the optimal coefficient vector. Finally, by considering some useful properties of the optimal coefficient vector, we modify the Schnorr-Euchner search algorithm to solve the SVP. We show that the estimated average complexity of our new algorithm is O(n1.5p0.5) flops for independent identically distributed (i.i.d.) Gaussian channel entries with SNR P based on the Gaussian heuristic. Simulations show that our algorithm is not only much more efficient than the existing ones that give the optimal solution, but also faster than some best known suboptimal methods. Besides, we show that our algorithm can be readily adapted to output a list of L best candidate vectors for use in the compute-and-forward design. The estimated average complexity of the resultant list-output algorithm is O(n2.5p0.5+ n1.5p0.5log(L) + nL) flops for i.i.d. Gaussian channel entries.

We consider the compute-and-forward protocol design problem with the objective being maximizing the computation rate at a single relay, and propose an efficient method that finds the optimal solution based on sphere decoding. The problem can be transformed into a shortest vector problem (SVP), which can be solved in two steps. First, by fully exploiting the specific structure of the associated Gram matrix using the hyperbolic transformation, the Cholesky factor can be computed with only n2/2 + O(n) flops. Then, taking into account of some useful properties of the optimal solution, we modify the Schnorr-Euchner search algorithm to solve the SVP. Numerical results show that our proposed branch-and-bound method is much more efficient than the existing one that gives the optimal solution. Besides, compared with the suboptimal methods, our method offers the best performance at a cost lower than that of the LLL based method and similar to that of the quadratic programming relaxation method.

The Korkine-Zolotareff (KZ) reduction has been used in communications and cryptography. In this paper, we modify a very recent KZ reduction algorithm proposed by Zhang et al., resulting in a new algorithm, which can be much faster and more numerically reliable, especially when the basis matrix is ill conditioned.

In this paper, cooperative localization of mobile nodes in non-line of sight (NLOS) situation is considered using a constrained square root unscented Kalman filter (CSRUKF). The NLOS measurements are used as quadratic constraints, which form a convex feasible region inside which the positions of the mobile nodes are supposed to be. The CSRUKF consists of two main stages: square root unscented Kalman filter (SRUKF) and sigma point projection. In the former, a conventional SRUKF is used to estimate the state vector and the Cholesky factor of the error covariance matrix. In the latter, a new set of sigma points are generated, and the ones violating the constraints are projected onto the feasible region by solving a set of convex quadratically constrained quadratic programs (QCQP). Each QCQP can be solved independently and in parallel for each sigma point violating the constraint, thus the algorithm is suitable for distributed processing. The simulation results show that our algorithm can perform well in different NLOS scenarios.

A common method to estimate an unknown integer parameter vector in a linear model is to solve an integer least squares (ILS) problem. A typical approach to solving an ILS problem is sphere decoding. To make a sphere decoder faster, the well-known LLL reduction is often used as preprocessing. The Babai point produced by the Babai nearest plane algorithm is a suboptimal solution of the ILS problem. First, we prove that the success probability of the Babai point as a lower bound on the success probability of the ILS estimator is sharper than the lower bound given by Hassibi and Boyd [1]. Then, we show rigorously that applying the LLL reduction algorithm will increase the success probability of the Babai point and give some theoretical and numerical test results. We give examples to show that unlike LLL's column permutation strategy, two often used column permutation strategies SQRD and V-BLAST may decrease the success probability of the Babai point. Finally, we show rigorously that applying the LLL reduction algorithm will also reduce the computational complexity of sphere decoders, which is measured approximately by the number of nodes in the search tree in the literature.

To efficiently utilize the computing resources and provide good quality of service (QoS) to the end-to-end tasks in the distributed real-time systems, we can enforce the utilization bounds on multiple processors. The utilization control is challenging especially when the workload in the system is unpredictable. To handle the workload uncertainties, current research favors feedback control techniques, and recent work combines the task rate adaptation and processor frequency scaling in an asynchronous way for CPU utilization control, where task rates and the processor frequencies are tuned asynchronously in two decoupled control loops for control convenience. Since the two manipulated variables, task rates and processor frequencies, contribute to the CPU utilizations together with strong coupling, adjusting them asynchronously may degrade the utilization control performance. In this paper, we provide a novel scheme to make synchronous rate and frequency adjustment to enforce the utilization setpoint, referred to as SyRaFa scheme. SyRaFa can handle the workload uncertainties by identifying the system model online and can simultaneously adjust the manipulated variables by solving an optimization problem in each sampling period. Extensive evaluation results demonstrate SyRaFa outperforms the existing schemes especially under severe workload uncertainties.

To provide satisfactory customer experience, modern server clusters like Amazon usually set Service Level Agreement (SLA) as guaranteeing a certain percentile (i.e. 99%) of the customer requests to have a response time within a threshold (i.e. 1s). One way to meet the SLA constraint is to serve the customer requests with sufficient computing capacity based on the worst case workload estimation in the server cluster. However, this may cause unnecessary power consumption in the server cluster due to over-provision of the computing capacity especially when the workload is highly dynamic. In this paper, we propose an adaptive computing capacity allocation scheme referred to as TailCon. TailCon aims at minimizing the power consumption in the server cluster while satisfying the SLA constraint by adjusting the number of active servers and the CPU frequencies of the turn on machines online. In TailCon, we analyze the distribution of the request response time dynamically and leverage the measured request response time to estimate the workload intensity in the server cluster, which is used as a continuous feedback to find the proper provision of the computing capacity online based on optimization techniques. We conduct both the emulation using the real-word HTTP traces and the experiments to evaluate the performance of TailCon. The experimental results demonstrate the effectiveness of TailCon scheme in enforcing the SLA constraint while saving the power consumption.

The maximum likelihood detection problem in many underdetermined linear communications systems can be described as an underdetermined integer least squares (ILS) problem. To solve it efficiently, a partial regularisation approach is proposed. The original underdetermined ILS problem is first transformed to an equivalent overdetermined ILS problem by using part of the transmit vector to do the regularisation. Then the overdetermined ILS problem is solved by conventional sphere decoding algorithms. Simulation results indicate that this approach can be much more efficient than other approaches for any square constellation higher than 4QAM.

An efficient maximum-likelihood decoder with a new reduction strategy is proposed for linear MIMO channel systems. Unlike the current reduction strategies which only reorder the columns of the channel matrix, the new reduction algorithm employs the so called integer Gauss transformations to reduce the off-diagonal entries of the upper triangular factor of the QR decomposition of the channel matrix. Simulation results show that this new decoding algorithm can be much more efficient than existing algorithms.

A box-constrained integer least squares problem (BILS) arises from several wireless communications applications. Solving a BILS problem usually has two stages: reduction (or preprocessing) and search. This paper presents a reduction algorithm and a search algorithm. Unlike the typical reduction algorithms, which use only the information of the lattice generator matrix, the new reduction algorithm also uses the information of the given input vector and the box constraint and is very effective for search. The new search algorithm overcomes some shortcomings of the existing search algorithms and gives some other improvement. Simulation results indicate the combination of the new reduction algorithm and the new search algorithm can be much more efficient than the existing algorithms, in particular when the least squares residual is large.

An efficient tree search decoder for underdetermined MIMO systems is presented. The decoder employs a novel column reordering strategy for the channel matrix in the reduction process, which can significantly reduce the computational cost. Simulation results show that this new algorithm is much more efficient than current approaches for a square constellation higher than 4QAM.

An efficient regularization approach is proposed for decoding underdetermined multiple input multiple output (MIMO) systems. The main idea is to transform an underdetermined integer least squares problem to an equivalent overdetermined integer least squares problem by using part of the transmit vector to do a regularization. Some strategies are proposed to enhance the efficiency of this approach. Specifically, we discuss how many entries of the transmit vector should be chosen and how to choose them when we do the regularization. An empirical formula for the regularization parameter is presented. Simulation results indicate that this modified approach can be much more efficient than current approaches for any square constellation higher than 4QAM.